mathematics Article On Killing Vector Fields on Riemannian Manifolds Sharief Deshmukh 1 and Olga Belova 2,* 1 Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia; [email protected] 2 Institute of Physical and Mathematical Sciences and IT, Immanuel Kant Baltic Federal University, A. Nevsky Str. 14, 236016 Kaliningrad, Russia * Correspondence: [email protected]; Tel.: +7-921-610-5949 Abstract: We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M, g) we show that for each non-constant smooth function f 2 C¥(M) there exists a non-zero vector field w f associated with f . In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M, g) with an appropriate lower bound on the integral of the Ricci curvature S(w f , w f ) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M, g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying krwk2 ≤ (n − 1)c and Ricci curvature in the direction of the vector field r f − w is bounded below by (n − 1)c is necessary and sufficient for (M, g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M, g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M, g) becomes a K-contact manifold. We also show that if in addition (M, g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M, g) is an Einstein Sasakian manifold. Keywords: killing vector field; K-contact manifold; sasakian manifold; Einstein–Sasakian manifold Citation: Deshmukh, S.; Belova, O. On Killing Vector Fields on Riemannian Manifolds. Mathematics 2021, 9, 259. https://doi.org/ 1. Introduction 10.3390/math9030259 Killing vector fields are known to play vital role in influencing the geometry as well Academic Editor: Pablo Sebastián as topology of Riemannian manifolds (see [1–10]) and being incompressible fields play Alegre Rueda important role in physics (see [11]). If we restrict the length of a Killing vector fields such Received: 14 January 2021 as constant length, then it severely restricts the geometry of Riemannian manifolds on Accepted: 25 January 2021 which they are set. For instance, there are no unit Killing vector fields on even-dimensional 2n Published: 28 January 2021 spheres S (c). However, there are unit Killing vector fields on odd-dimensional spheres S2n+1(c) as well on odd-dimensional ellipsoids (see [4,12,13]). Most importantly, on all K- Publisher’s Note: MDPI stays neu- contact manifolds there is a unit Killing vector field called the Reeb vector field (see [12,13]). tral with regard to jurisdictional clai- There are other important structures and special vector fields, which also influence the ms in published maps and institutio- geometry of a Riemannian manifold (see [14]). nal affiliations. In this paper, we are interested in studying the impact of presence of a unit Killing vector field w on the geometry of a connected Riemannian manifold (M, g). Our first interesting finding is that for each smooth non-constant function f on M, there is naturally associated a non-zero vector field w f on M that is orthogonal to w (see Proposition 1). Then Copyright: © 2021 by the authors. Li- w f f censee MDPI, Basel, Switzerland. we consider the associated vector field corresponding to eigenfunction of Laplace ( ) This article is an open access article operator on a compact Riemannian manifold M, g corresponding to eigenvalue dim M f distributed under the terms and con- and show that if the integral of Ricci curvature in the direction of w is bounded below ditions of the Creative Commons At- by certain bound, forces dim M to be odd and thus gives a necessary and sufficient for 2m+1 tribution (CC BY) license (https:// (M, g) to be isometric to the unit sphere S (see Theorem 1). A similar characterization 2m+1 creativecommons.org/licenses/by/ of the sphere S (c) using the eigenfunction f on a compact n-dimensional (M, g) with 4.0/). eigenvalue nc is obtained (see Theorem 2). Mathematics 2021, 9, 259. https://doi.org/10.3390/math9030259 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 259 2 of 17 Given a unit Killing vector field w on a connected Riemannian manifold (M, g) there is associated skew-symmetric tensor field Y given by the covariant derivative of w, giving a structure (Y, w, a, g) on M, where a is 1-form dual to w. A natural question is under which situation the structure (Y, w, a, g) becomes a K-contact structure? We answer this question and find a necessary and sufficient condition in terms of sectional curvatures of plane sections containing w (see Theorem 3). We also find conditions under which a Riemannian manifold admitting a unit Killing vector field is an Einstein manifold (see Theorems 4 and 5). Finally, as a by-product of Theorem 3 with an additional condition that the Ricci operator is Codazzi type tensor on a Riemannian manifold (M, g) that admits a unit Killing vector field w, we show that (M, g) is an Einstein Sasakian manifold (see Corollary 2). 2. Preliminaries Recall that a vector field w on a Riemannian manifold (M, g) is said to be a Killing vector fields if £wg = 0, where £w being Lie-derivative of metric g with respect to x; or equivalently, r = − r 2 X( ) g X1 w, X2 g X2 w, X1 , X1, X2 M , (1) where r is the Riemannian connection on (M, g) and X(M) is Lie-algebra of smooth vector fields on M. Please note that a parallel vector field is a Killing vector field, we say a Killing vector field is a non-trivial Killing vector field if it is not parallel. For the Killing vector field w on an n-dimensional Riemannian manifold (M, g), we denote by a the smooth 1-form dual to w, i.e., a(X) = g(w, X), X 2 X(M). In addition, we define a skew-symmetric (1.1)-tensor field Y by 1 1 1 g(YX , X ) = da(X , X ) = gr w, X − gr w, X , X , X 2 X(M). 1 2 2 1 2 2 X1 2 2 X2 1 1 2 Then, using Equation (1), it follows that r = 2 X( ) g X1 w, X2 g(YX1, X2), X1, X2 M , that is, rXw = YX, X 2 X(M). (2) The curvature tensor field R and the Ricci tensor S of (M, g) are given by R(X X )X = r r X − r X 1, 2 3 X1 , X2 3 [X1,X2] 3 (3) and n S(X1, X2) = ∑ g(R(ei, X1)X2, ei), i=1 respectively, where fe1, .., eng is a local orthonormal frame on M. According to symmetry of the Ricci tensor S, we get a symmetric operator T called the Ricci operator of M and defined by g(TX1, X2) = S(X1, X2), X1, X2 2 X(M). The scalar curvature t of M is given by t = Tr T. Please note that rt—the gradient of the scalar curvature t—satisfies 1 n rt = ∑(rT)(ei, ei), (4) 2 i=1 Mathematics 2021, 9, 259 3 of 17 where rT is the covariant derivative of T, defined by r ( ) = r − r ( T) X1, X2 X1 TX2 T X1 X2. Please note that if w is a Killing vector field on a Riemannian manifold (M, g), then using Equations (2) and (3), we have R(X1, X2)w = (rY)(X1, X2) − (rY)(X2, X1), X1, X2 2 X(M). (5) From definition of Y, we see that the smooth 2-form g(YX1, X2) is closed and as a result of using Equation (5), we make a conclusion (rY)(X1, X2) = R(X1, w)X2, X1, X2 2 X(M). (6) Observe that if w is a Killing vector field of constant length on a Riemannian manifold (M, g), then taking the inner product with w in Equation (2), we get g(YX, w) = 0, X 2 X(M), and as Y is skew-symmetric, we conclude Yw = 0. (7) For given a smooth function f on a Riemannian manifold the Hessian operator H f is defined by H f (X) = rXr f , X 2 X(M), (8) which is a symmetric operator with TrH f = D f , where D is the Laplace operator. The Hessian Hess( f ) of f is defined by Hess( f )(X1, X2) = g H f (X1), X2 , X1, X2 2 X(M). Let w be a Killing vector field on a Riemannian manifold (M, g) and C¥(M) be the algebra of smooth functions on M. Then, using Equation (2) and skew-symmetry of the operator Y, it follows that divw = 0 and that for a smooth function f 2 C¥(M), we have div( f w) = w( f ). Thus, we get Lemma 1. Let w be a unit Killing vector field on a compact Riemannian manifold (M, g). Then for a smooth function f 2 C¥(M) Z w( f ) = 0.